Partially dissipative hyperbolic systems in the critical regularity setting : the multi-dimensional case

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Partially dissipative hyperbolic systems in the critical regularity setting : the multi-dimensional case

Timothée Crin-Barat, Raphaël Danchin

To cite this version:

Timothée Crin-Barat, Raphaël Danchin. Partially dissipative hyperbolic systems in the critical regu-

larity setting : the multi-dimensional case. 2021. �hal-03223784�

PARTIALLY DISSIPATIVE HYPERBOLIC SYSTEMS IN THE CRITICAL REGULARITY SETTING : THE MULTI-DIMENSIONAL CASE

TIMOTHÉE CRIN-BARAT, RAPHAËL DANCHIN

Abstract. We are concerned with quasilinear symmetrizable partially dissipative hyperbolic systems in the whole spaceR^d with d ≥2.Following our recent work [10] dedicated to the one-dimensional case, we establish the existence of global strong solutions and decay estimates in the critical regularity setting whenever the system under consideration satisfies the so-called (SK) (for Shizuta-Kawashima) condition. Our results in particular apply to the compressible Euler system with damping in the velocity equation.

Compared to the papers by Kawashima and Xu [27, 28] devoted to similar issues, our use ofhybridBesov norms with different regularity exponents in low and high frequency enable us to pinpoint optimal smallness conditions for global well-posedness and to get more accurate information on the qualitative properties of the constructed solutions.

A great part of our analysis relies on the study of a Lyapunov functional in the spirit of that of Beauchard and Zuazua in [2]. Exhibiting a damped mode with faster time decay than the whole solution also plays a key role.

Introduction

We are concerned with first order n-component systems in R

^d

of the type:

(1) A

⁰

(V ) ∂V

∂t + X

d

j=1

A

^j

(V ) ∂V

∂x

_j

= H(V )

where the (smooth) matrices valued functions A

^j

(j = 0, · · · , d) and vector valued function H are defined on some open subset O

V

of R

ⁿ

and the unknown V = V (t, x) depends on the time variable t ∈ R

₊

and on the space variable x ∈ R

^d

(d ≥ 2). We assume that the system is symmetrizable and satisfies additional structure assumptions that will be specified in the next section.

System (1) is supplemented with initial data V

₀

∈ O

V

at time t = 0. We are concerned with the existence of global strong solutions in the case where V

₀

is close to some constant state V ¯ such that H( ¯ V ) = 0.

In the nondissipative case, that is if H ≡ 0, it is classical that symmetrizable quasilinear hyperbolic systems supplemented with initial data with Sobolev regularity H

^s

such that s >

1+d/2 admit local-in-time strong solutions (see e.g. [3]), that may develop singularities in finite time even if the initial data are small (see for instance the works by Majda in [18] or Serre in [21]). By contrast, if in the neighborhood of V , ¯ the term H(V ) has the ‘good’ sign and acts on each component of the solution (like e.g. H(V ) = D(V − V ¯ ) for some matrix D having all its eigenvalues with positive real part), then smooth perturbations of V ¯ give rise to global-in-time solutions that tend exponentially fast to V ¯ when time goes to ∞ .

In most physical situations that may be modelled by systems of the form (1) however, some components of the solution satisfy conservation laws and only partial dissipation occurs, that is to say, the term H(V ) acts only on a part of the solution. Typically, this happens in gas dynamics where the mass density and entropy are conserved, or in numerical schemes involving conservation laws with relaxation. A well known example is the damped compressible Euler system for isentropic flows that will be addressed at the end of the paper. For this system, it is known from the works of Wang and Tang [25] or Sideris, Thomases and Wang [23] that the

1

dissipative mechanism, albeit only present in the velocity equation, can prevent the formation of singularities that would occur for H ≡ 0.

Looking for sufficient conditions on the dissipation term H guaranteeing the global ex- istence of strong solutions for perturbations of a constant state V ¯ goes back to the thesis of Kawashima [15] and to the more recent work by Yong in [29]. Two main conditions arise. The first one is the so-called (SK)(for Shizuta-Kawashima) stability condition, see [22], that ensures that the damping is strong enough to prevent the solutions emanating from small perturbations of V ¯ from blowing up. The second one is the existence of a (dissipative) entropy which provides a suitable symmetrisation of the system compatible with H. Thanks to those two conditions, Yong [29] obtained a global existence result for systems that are more general than those that have been considered by Kawashima.

More recently, by taking advantage of the properties of the Green kernel of the linearized system around V ¯ and on the Duhamel formula, Bianchini, Hanouzet and Natalini in [5] pointed out the convergence in L

^p

of global solutions to V , ¯ with the rate O (t

^{−d2(1−1p)}

) when t → + ∞ , for all p ∈ [min { d, 2 } , ∞ ]. Let us further mention that Kawashima and Yong proved decay estimates in regular Sobolev space in [17].

A few years ago, Kawashima and Xu in [27] and [28] extended the prior works on par- tially dissipative hyperbolic systems satisfying the (SK) and entropy conditions to critical non- homogeneous Besov spaces. To obtain their results, they used the symmetrisation from [16]

and applied a frequency localisation argument relying on the Littlewood-Paley decomposition.

In their work, the equivalence between Condition (SK) and the existence of a compensating function allows to exhibit the global-in-time L

²

integrability properties of all the components of the solution.

However, it is known that the condition (SK) is not optimal in the sense that there exist many systems that do not verify it but for which one can prove global well-posedness results, see e.g. [20, 4, 14]. In [2], Beauchard and Zuazua developed a new and systematic approach that allows to establish global existence results and to describe large time behavior of solutions to partially dissipative systems that need not satisfy Condition (SK). Looking at the linearization of System (1) around a constant solution, namely (denoting from now on ∂

_t

,

_∂t^∂

and ∂

_j

,

_∂x^∂

j

),

(2) ∂

_t

Z +

X

m

j=1

A

^j

∂

_j

Z = − LZ,

they show that Condition (SK) is equivalent to the Kalman maximal rank condition on the matrices A

^j

and L. More importantly, they introduce a Lyapunov functional equivalent to the L

²

norm that encodes enough information to recover dissipative properties of (2). Considering such a functional is motivated by the classical (linear) control theory of ODEs, and is also related to Villani’s paper [24]. Back to the nonlinear system (1), Beauchard and Zuazua obtained the existence of global smooth solutions for perturbations of a constant equilibrium V ¯ that satisfies (SK) Condition. Furthermore, using arguments borrowed from Coron’s return method [9], they were able to achieve certain cases where (SK) does not hold.

Our aim here is to extend the results we obtained recently in the one-dimensional case [10]

to multi-dimensional partially dissipative hyperbolic systems (see also the on-going work [6] by

the first author dedicated to the relaxation limit of a non conservative multi-fluid system that

does not satisfy the (SK) condition). More precisely, under Condition (SK), we shall develop

Beauchard and Zuazua’s approach as suggested by the second author in [12] and prove the

global well-posedness of (1) supplemented with data that are close to V ¯ in an optimal critical

regularity setting. As in the study of the compressible Navier-Stokes system and related models

(see e.g. [7, 8, 11, 13]) it will appear naturally that in order to get optimal results, one has

to use functional spaces with different regularity exponents in low and high frequencies. Here,

Beauchard and Zuazua’s approach will give us the information that the low frequencies (resp.

high frequencies) of the solution of the linearized system behave like the heat flow (resp. are exponentially damped). Furthermore, in order to improve our low frequency analysis, we will exhibit a damped mode with better decay properties than the whole solution. Thanks to that, we will end up with more accurate estimates and a weaker smallness condition that in prior works (in particular [27]) and refine the decay estimates that were obtained in [28].

The paper is arranged as follows. In the first section, we specify the structure of the class of partially dissipative hyperbolic systems we aim at considering, and explain the construction of a Lyapunov functional that will be the key to our global results. In Section 2, we state the main results of the paper. Section 3 is devoted to the proof of a first global existence result and time decay estimates for general partially dissipative systems satisfying the Shizuta- Kawashima condition. In section 4, under additional structure assumptions (that are satisfied by the compressible Euler system with damping), we obtain a more accurate global existence result. Some technical results are proved or recalled in Appendix.

1. Hypotheses and method

In this section, we specify our assumptions on the system under consideration, and explain the main steps of our approach.

1.1. Friedrichs-symmetrizability. First, we fix some constant solution V ¯ ∈ O

V

of (1) (thus satisfying H( ¯ V ) = 0). To ensure the local well-posedness, we assume that (1) is Friedrichs- symmetrizable, namely that there exists a smooth function S : V 7→ S(V ) defined on O

V

, valued in the set of symmetric and positively definite matrices such that for all V ∈ O

V

, the matrices (SA

⁰

)(V ), · · · , (SA

^d

)(V ) are symmetric and, in addition, (SA

⁰

(V )) is definite positive.

Denoting H e , SH and A e

^j

, SA

^j

for j ∈ { 0, · · · , } , System (1) rewrites A e

⁰

(V )∂

_t

V +

X

d

j=1

A e

^j

(V )∂

_j

V = H(V e ).

Then, setting Z , V − V , L ¯ , − D

_V

H( ¯ e V ) and r(Z) , H( ¯ e V + Z) + LZ, we get

(3) A e

⁰

(V )∂

t

Z +

X

d

j=1

A e

^j

(V )∂

j

Z + L Z = r(Z).

By construction, the remainder r is at least quadratic with respect to Z.

Second, we assume that System (1) is partially dissipative in the following meaning:

(i) The whole space R

ⁿ

may be decomposed into R

ⁿ

= M L

M

^⊥

where M =

φ ∈ R

ⁿ

, h φ, H(V e ) i = 0 for all V ∈ O

V

}·

Hence, denoting by P the orthogonal projection on M , we may write

(4) V =

V

₁

V

2

and H(V ) = 0

H

2

(V )

where V

₁

= P V ∈ R

ⁿ¹

, V

₂

= (I − P )V ∈ R

ⁿ²

and n

₁

+ n

₂

= n.

(ii) The linear map L , − D

_V

H( ¯ e V ) is an isomorphism on M

^⊥

such that for some c > 0, (5) ∀ η ∈ R

ⁿ

, (Lη | η) ≥ c | Lη |

²

.

(iii) System (1) has a block structure that is compatible with decomposition (4), namely all

the matrices A e

^j

are diagonal by blocks (first block being of size n

₁

× n

₁

and second one

of size n

₂

× n

₂

) and we have r(Z

₁

, 0) = 0 for all Z

₁

close to 0. This entails that r is at

least linear with respect to Z

2

.

According to the above assumptions and introducing the decompositions:

A e

⁰

= A e

⁰_1,1

0 0 A e

⁰_2,2

!

, A e

^j

= A e

^j_1,1

A e

^j_1,2

A e

^j_2,1

A e

^j_2,2

!

, L = 0

L

and r = 0

Q

, System (3) may thus be rewritten as:

(6)

 

 

 



 

 

 

A e

⁰_1,1

(V )∂

_t

Z

₁

+ X

d

j=1

A e

^j_1,1

(V )∂

_j

Z

₁

+ A e

^j_1,2

(V )∂

_j

Z

₂

= 0, A e

⁰_2,2

(V )∂

_t

Z

₂

+

X

d

j=1

A e

^j_2,1

(V )∂

_j

Z

₁

+ A e

^j_2,2

(V )∂

_j

Z

₂

+ L Z

₂

= Q(Z ).

As we shall see in Section 4, the compressible Euler equations with damping, rewritten in suitable variables, satisfies the above assumptions about any constant state with positive density and null velocity.

1.2. The Shizuta-Kawashima and Kalman rank conditions. In order to specify the sup- plementary conditions on the structure of the system ensuring global well-posedness and present the overall strategy, let us consider the linearization of (1) about V , ¯ namely:

(7) A ¯

⁰

∂

_t

Z + X

d

j=1

A ¯

^j

∂

_j

Z + LZ = G with A ¯

^j

:= A e

^j

( ¯ V ) for j = 0, · · · , d.

Then, owing to the symmetry of the matrices A ¯

^j

, the classical energy method leads to

(8) 1

2 d dt k Z k

²_L²_¯

A0

+ (LZ | Z ) = 0 with k Z k

²_L²_¯

A0

, ( ¯ A

₀

Z | Z ).

On the one hand, since the matrix A ¯

0

is symmetric and positive definite, we have

(9) k Z k

L²_A¯

0

≃ k Z k

²L²

.

On the other hand, (5) and the definition of Z

₂

guarantee that there exists κ

₀

> 0 such that (10) (LZ | Z ) ≥ κ

₀

k Z

₂

k

²L²

for all Z ∈ L

²

( R

^d

; R

ⁿ

).

Hence, (8) yields L

²

-in-time integrability on the components of Z experiencing direct dissipa- tion, but not on the whole solution. To compensate this lack of coercivity, following Beauchard and Zuazua in [2], we are going to introduce a lower order corrector I to track the optimal dis- sipation of the solution to (7). Since it is more natural to define that corrector on the Fourier side, let us look at (7) in the Fourier space, that is, denoting by ξ ∈ R

^d

the Fourier variable,

A ¯

⁰

∂

_t

Z b + i X

d

j=1

A ¯

^j

ξ

_j

Z b + L Z b = G. b

Let us write ξ = ρω with ω ∈ S

^d−1

and ρ = | ξ | . Then, the above system rewrites (11) ∂

_t

Z b + iρM

_ω

Z b + N Z b = ¯ A

⁻¹₀

G b with M

_ω

, A ¯

⁻¹₀

X

d

j=1

ω

_j

A ¯

^j

and N , A ¯

⁻¹₀

L.

Clearly, since A ¯

⁻¹₀

is positive definite, (5) implies that there exists a positive constant (still denoted by κ

₀

) so that

(12) ∀ η ∈ R

ⁿ

, (N η | η) ≥ κ

0

| N η |

²

.

Fix n − 1 positive parameters ε

₁

, · · · ε

_n−1

(bound to be small), and set

(13) I , ℜ

n−1

X

k=1

ε

_k

N M

_ω^k−1

Z b · N M

_ω^k

Z b where · designates the Hermitian scalar product in C

ⁿ

.

For expository purpose, assume that G ≡ 0. Then, differentiating I with respect to time and using (11) yields

(14) d dt I +

n−1

X

k=1

ε

_k

ρ | N M

_ω^k

Z b |

²

= −ℑ

n−1

X

k=1

ε

_k

N M

_ω^k−1

N Z b · N M

_ω^k

Z b

+ ℜ

n−1

X

k=1

ε

_k

ρ N M

_ω^k−1

Z b · N M

_ω^k+1

Z b

− ℑ

n−1

X

k=1

ε

_k

N M

_ω^k−1

Z b · N M

_ω^k

N Z b

· As pointed out in [2] (and recalled in Appendix for the reader’s convenience), it is possible to choose positive and arbitrarily small parameters ε

1

, · · · , ε

n−1

so that (14) implies for some C > 0,

(15) d

dt I + 1 2

n−1

X

k=1

ε

_k

ρ | N M

_ω^k

Z b |

²

≤ κ

₀

2(2π)

^d

ρ | N Z b |

²

+ Cε

₁

| N Z b |

²

.

Setting ε

₀

= (2π)

^−d

κ

₀

/2, taking ε

₁

small enough, integrating on R

^d

, using Fourier-Plancherel theorem and combining with (8), we end up with

(16) d

dt L + H ≤ 0 with H , Z

R^d n−1

X

k=0

ε

_k

min(1, | ξ |

²

) | N M

_ω^k

Z b (ξ) |

²

dξ and L , k Z k

²_L²_¯

A0

+ Z

R^d

min( | ξ | , | ξ |

⁻¹

) I (ξ) dξ.

Clearly, if ε

₁

, · · · , ε

_n−1

are small enough, then L ≃ k Z k

²_L2

. The question now is whether H may be compared to k Z k

²_L²

. The answer depends on the properties of the support of Z b

₀

and on the possible cancellation of the following quantity:

(17) N

V^¯

:= inf

n−1

X

k=0

ε

_k

| N M

_ω^k

x |

²

; x ∈ S

ⁿ⁻¹

, ω ∈ S

^d−1

·

At this very point, the (SK) (for Shizuta and Kawashima) condition comes into play:

Definition 1.1. System (1) verifies the (SK) condition at V ¯ ∈ M if, for all ω ∈ S

^d−1

, whenever φ ∈ R

ⁿ

satisfies N φ = 0 and λφ + M

_ω

φ = 0 for some λ ∈ R , we must have φ = 0.

It is clear that Condition (SK) at V ¯ is equivalent to:

∀ ω ∈ S

^d−1

, ker N ∩ { eigenvectors of M

_ω

} = { 0 } .

In order to pursue our analysis, we need the following key result (see the proof in e.g. [2]).

Proposition 1.1. Let M and N be two matrices in M

n

( R ). The following assertions are equivalent:

(1) N φ = 0 and λφ + M φ = 0 for some λ ∈ R implies φ = 0;

(2) For every ε

₀

, · · · , ε

_n−1

> 0, the function y 7−→

v u u t

n−1

X

k=0

ε

_k

| N M

^k

y |

²

defines a norm on R

ⁿ

;

Thanks to the above proposition and observing that the unit sphere S

^d−1

is compact, one may conclude that Condition (SK) is satisfied by the pair (M

ω

, N ) for all ω ∈ S

^d−1

if and only if N

V^¯

> 0. Furthermore, we note that:

• if Z b

₀

is compactly supported then, H & k∇ Z k

²_L2

, which reveals a parabolic behavior of all components of the solution;

• if the support of Z b

₀

is away from the origin, then H & k Z k

²_L2

, which corresponds to exponential decay.

Therefore, at the linear level, in order to get optimal dissipative estimates, it is suitable to split the solution into low and high frequencies parts. This will actually be achieved by means of a Littlewood-Paley decomposition (introduced in the next section). Then, a great part of our analysis will consist in localizing (3) on the Fourier side by means of this decomposition, and to study the evolution of the functional L pertaining to each part.

1.3. The damped mode. Another important ingredient of our analysis is the use of a ‘damped mode’ that, somehow, may be seen as an eigenmode corresponding to the part of the solution that experiences maximal dissipation in low frequencies. It is defined as follows :

(18) W , −L

⁻¹

A e

⁰_2,2

(V )∂

_t

Z

₂

= Z

₂

+ X

d

j=1

L

⁻¹

A e

^j_2,1

(V )∂

_j

Z

₁

+ A e

^j_2,2

(V )∂

_j

Z

₂

− L

⁻¹

Q(Z) · Note that

(19) A e

⁰_2,2

(V )∂

_t

W + L W = A e

⁰_2,2

(V ) L

⁻¹

X

d

j=1

∂

_t

A e

^j_2,1

(V )∂

_j

Z

₁

+ A e

^j_2,2

(V )∂

_j

Z

₂

− A e

⁰_2,2

(V ) L

⁻¹

∂

_t

Q(Z ) · On the left-hand side, Property (5) ensures maximal dissipation on W. As the right-hand side of (19) contains only at least quadratic terms, or linear terms with one derivative, it can be expected to be negligible in low frequencies if Z is small enough. Furthermore, (18) reveals that W is comparable to Z

2

in low frequencies. This will ensure better integrability for Z

2

than for the whole solution Z.

2. Main results

Before stating our main results, introducing a few notations is in order.

First, we fix a homogeneous Littlewood-Paley decomposition ( ˙ ∆

_q

)

_q∈Z

that is defined by

∆ ˙

_q

, ϕ(2

^−q

D) with ϕ(ξ) , χ(ξ/2) − χ(ξ)

where χ stands for a smooth function with range in [0, 1], supported in the open ball B(0, 4/3) and such that χ ≡ 1 on the closed ball B(0, ¯ 3/4). We further state

S ˙

_q

, χ(2

^−q

D) for all q ∈ Z and define S

_h^′

to be the set of tempered distributions z such that

q→−∞

lim k S ˙

_q

z k

L^∞

= 0.

Following [1], we introduce the homogeneous Besov semi-norms:

k z k

B˙^s

p,r

, 2

^qs

k ∆ ˙

_q

z k

L^p(R^d)

ℓ^r(Z)

,

then define the homogeneous Besov spaces B ˙

^s_p,r

(for any s ∈ R and (p, r) ∈ [1, ∞ ]

²

) to be the

subset of z in S

_h^′

such that k z k

B˙^s_p,r

is finite.

Using from now on the shorthand notation

(20) ∆ ˙

_q

z , z

_q

,

we associate to any element z of S

h^′

, its low and high frequency parts through z

^ℓ

, X

q≤0

z

q

= ˙ S

1

z and z

^h

, X

q>0

z

q

= (Id − S ˙

1

)z.

We shall constantly use the following Besov semi-norms for low and high frequencies:

k z k

^ℓB˙^s

2,1

, X

q≤0

2

^qs

k z

q

k

L²

and k z k

^hB˙^s

2,1

, X

q>0

2

^qs

k z

q

k

L²

, k z k

^ℓB˙^s

2,∞

, sup

q≤0

2

^qs

k z

_q

k

L²

and k z k

^hB˙^s

2,∞

, sup

q>0

2

^qs

k z

_q

k

L²

. Throughout the paper, we shall use repeatedly the following obvious fact:

(21) k z k

^ℓB˙^s_2,r^′

≤ k z k

^ℓB˙^s

2,r

and k z k

^hB˙^s_2,r^′

≥ k z k

^hB˙^s

2,r

for r = 1, ∞ , whenever s ≤ s

^′

.

For any Banach space X, index ρ in [1, ∞ ] and time T ∈ [0, ∞ ], we use the notation k z k

L^ρ_T(X)

, k z k

X

L^ρ(0,T)

. If T = + ∞ , then we just write k z k

L^ρ(X)

. Finally, in the case where z has n components z

_j

in X, we keep the notation k z k

_X

to mean P

j∈{1,···,n}

k z

_j

k

_X

. We can now state our main global existence result for System (1), rewritten as (3).

Theorem 2.1. Let V ¯ be an equilibrium state such that H( ¯ V ) = 0 and suppose that the structure assumptions of paragraph 1.1 and (SK) condition are satisfied. Then, there exists a positive constant α such that for all Z

₀

∈ B ˙

d 2−1 2,1

∩ B ˙

d 2+1

2,1

satisfying

(22) Z

0

, k Z

₀

k

^ℓ

˙ B

d 2−1 2,1

+ k Z

₀

k

^h

˙ B

d 2,12 +1

≤ α,

System (3) supplemented with initial data Z

₀

admits a unique global-in-time solution Z in the space E defined by

Z ∈ C

b

( R

⁺

; ˙ B

d 2−1 2,1

∩ B ˙

d 2+1

2,1

), Z

^h

∈ L

¹

( R

⁺

; ˙ B

d 2+1

2,1

), Z

₁^ℓ

∈ L

¹

( R

⁺

; ˙ B

d 2+1

2,1

) and W ∈ L

¹

( R

⁺

; ˙ B

d 2−1 2,1

), with W defined according to (18).

Moreover, there exists a Lyapunov functional that is equivalent to k Z k

_˙

B

d2−1 2,1 ∩B˙

d2 +1 2,1

, and a constant C depending only on the matrices A

^j

and on H, such that

(23) Z (t) ≤ C Z

0

for all t ≥ 0

where

(24) Z (t) , k Z k

^ℓ

L^∞_t ( ˙B

d2−1 2,1 )

+ k Z k

^h

L^∞_t ( ˙B

d2 +1 2,1 )

+ k Z k

L¹_t( ˙B

d2 +1 2,1 )

+ k W k

^ℓ

L¹_t( ˙B

d2−1 2,1 )

+ k Z

₂

k

^ℓ

L¹_t( ˙B

d2 2,1)

+ k Z

₂

k

^ℓ

L²_t( ˙B

d2−1 2,1 )

. Remark 2.1. As is, the above theorem does not extend to the case d = 1. The reason why is that the low frequency regularity index then becomes negative, so that some nonlinear terms cannot be bounded in the proper spaces. For more details, the reader may refer to [10].

Our second result concerns the time-decay estimates of the solution we constructed in

Theorem 2.1.

Theorem 2.2. Under the hypotheses of Theorem 2.1 and if, additionally, Z

0

∈ B ˙

^−σ1

2,∞

for some σ

₁

∈

−

^d₂

,

^d₂

then, there exists a constant C depending only on σ

₁

and such that

(25) k Z(t) k

B˙^−σ1

2,∞

≤ C k Z

₀

k

B˙^−σ1

2,∞

, ∀ t ≥ 0.

Furthermore, if σ

₁

> 1 − d/2 then, denoting h t i , p

1 + t

²

, α

₁

, σ

₁

+

^d₂

− 1

2 and C

₀

, k Z

₀

k

^ℓ_B˙^−σ1 2,∞

+ k Z

₀

k

^h

˙ B

d 2,12 +1

, we have the following decay estimates:

sup

t≥0

h t i

^σ+σ21

Z(t)

^ℓ_B˙σ 2,1

≤ CC

₀

if − σ

₁

< σ ≤ d/2 − 1, sup

t≥0

h t i

^σ+σ21+12

Z

2

(t)

^ℓ_B˙σ

2,1

≤ CC

0

if − σ

1

< σ ≤ d/2 − 2, sup

t≥0

kh t i

^α1

Z

₂

(t) k

^ℓB˙^σ

2,1

≤ CC

₀

if min(d/2 − 2, − σ

₁

) < σ ≤ d/2 − 1 and sup

t≥0

h t i

^2α1

Z(t)

^h

B˙

d 2,12 +1

≤ CC

₀

.

Remark 2.2. Since we have the embedding L

¹

֒ → B ˙

⁻

d

2,∞2

, the above statement encompasses the classical decay assumption Z

₀

∈ L

¹

(see e.g. [19] in a slightly different context).

Remark 2.3. Owing to the presence of "direct" dissipation in the equation of Z

₂

, the decay of the low frequencies of Z

₂

is stronger by a factor 1/2 than the decay of the whole solution.

If we assume in addition that:

(26)

 

 

 



For all j ∈ { 1, · · · , d } , A

^j_1,1

( ¯ V ) = 0 and D

_V1

A

^j_1,1

( ¯ V ) = 0;

For all j ∈ { 1, · · · , d } , D

_V1

A

^j_2,1

( ¯ V ) = 0 (and thus also D

_V1

A

^j_1,2

( ¯ V ) = 0);

The function r is quadratic with respect to Z

₂

(i.e. D

_V²_i,Vj

r(0) = 0 for (i, j) 6 = (2, 2)), then one can weaken the low frequency assumption, as we did in our work [10] dedicated to one-dimensional case, and get:

Theorem 2.3. Let the assumptions of Theorem 2.1 concerning system (1) be in force and assume in addition that (26) holds true.

Then, there exists a positive constant α such that for all Z

₀

∈ B ˙

d

2,12

∩ B ˙

d 2+1

2,1

satisfying

(27) Z

0^′

, k Z

0

k

^ℓ

˙ B

d2 2,1

+ k Z

0

k

^h

˙ B

d2 +1 2,1

≤ α,

System (3) supplemented with initial data Z

₀

admits a unique global-in-time solution Z in the space F defined by

Z ∈ C

b

( R

⁺

; ˙ B

d

2,12

∩ B ˙

d 2+1

2,1

), Z

^h

∈ L

¹

( R

⁺

; ˙ B

d 2+1

2,1

), Z

₁^ℓ

∈ L

¹

( R

⁺

; ˙ B

d 2+2

2,1

) and W ∈ L

¹

( R

⁺

; ˙ B

d

2,12

).

Moreover, there exists a Lyapunov functional that is equivalent to k Z k

_˙

B

d 2,12 ∩B˙

d 2,12 +1

, and we have the following a priori estimate:

(28) Z

^′

(t) ≤ C Z

0^′

where Z

^′

(t) , k Z k

^ℓ

L^∞_t ( ˙B

d

2,12 )

+ k Z k

^h

L^∞_t ( ˙B

d 2,12 +1)

+ k Z

₁

k

^ℓ

L¹_t( ˙B

d2 +2

2,1 )

+ k Z

₂

k

^ℓ

L¹_t( ˙B

d2 +1

2,1 )

+ k Z

₂

k

^ℓ

L²_t( ˙B

d2

2,1)

+ k Z k

^h

L¹_t( ˙B

d2 +1

2,1 )

+ k W k

^ℓ

L¹_t( ˙B

d2 2,1)

.

Finally, if, additionally, Z

0

∈ B ˙

^−σ1

2,∞

for some σ

1

∈

−

^d₂

,

^d₂

then (25) is satisfied as well as the decay estimates that follow, up to σ = d/2 for the first one, and with d/2 − 1 and d/2 instead of d/2 − 2 and d/2 − 1 for the next two ones, with α

₁

replaced by (σ

₁

+ d/2)/2.

Remark 2.4. As will be shown in the last section, this theorem applies to the compressible Euler with damping (see Theorem 4.1).

Remark 2.5. In contrast with Theorem 2.1, the functional setting of Theorem 2.3 allows to obtain uniform estimates in the asymptotic λ → + ∞ if the dissipative term is λH. This is the first step for studying the high relaxation limit.

3. Proof of Theorems 2.1 and 2.2

This section is devoted to proving the global existence of strong solutions and decay esti- mates for System (1) supplemented with initial data that are close to the reference solution V , ¯ in the general case where the structural assumptions listed in Subsection 1.1 and (SK) condition are satisfied.

The bulk of the proof consists in establishing a priori estimates, the other steps (proving existence and uniqueness) being more classical. As explained before, our strategy is to first work out a Lyapunov functional in Beauchard-Zuazua’s style, that is equivalent to the norm that we aim at controlling, then to combine with the study of the damped mode W defined in (18) so as to close the estimates.

3.1. Establishing the a priori estimates. Throughout this part, we assume that we are given a smooth (and decaying) solution Z of (3) on [0, T ] × R

^d

with Z

₀

as initial data, satisfying

(29) sup

t∈[0,T]

k Z (t) k

_˙

B

d 2,12

≪ 1.

We shall use repeatedly that, owing to the embedding B ˙

d

2,12

֒ → L

^∞

, we have also

(30) sup

t∈[0,T]

k Z(t) k

L^∞

≪ 1.

From now on, C > 0 designates a generic harmless constant, the value of which depends on the context and we denote by (c

q

)

q∈Z

nonnegative sequences such that P

q∈Z

c

q

= 1.

To start with, let us rewrite (3) as follows:

(31) A ¯

⁰

∂

t

Z +

X

d

j=1

A ¯

^j

∂

j

Z + LZ = G

with G , G

1

+ G

2

+ G

3

and G

₁

, −

X

d

j=1

A ¯

⁰

( A e

⁰

(V ))

⁻¹

A e

^j

(V ) − ( ¯ A

⁰

)

⁻¹

A ¯

^j

∂

_j

Z, G

2

, − A ¯

⁰

( A e

⁰

(V ))

⁻¹

− ( ¯ A

⁰

)

⁻¹

LZ, G

₃

, A ¯

⁰

( A e

⁰

(V ))

⁻¹

r(Z).

For q ∈ Z , applying ∆ ˙

_q

to (31) yields (32) A ¯

⁰

∂

_t

Z

_q

+

X

d

j=1

A ¯

^j

∂

_j

Z

_q

+ LZ

_q

= ˙ ∆

_q

G with Z

_q

, ∆ ˙

_q

Z.

Our analysis will mainly consist in estimating for all q ∈ Z a functional L

q

that is equivalent to the L

²

( R

^d

; R

ⁿ

) norm of Z

_q

and encodes informations on the dissipative properties of the system.

That functional will be built from (16) and, since Condition (SK) is satisfied, the number N

V^¯

defined in (17) will be positive. Furthermore, since the Fourier transform of Z

_q

is localized near the frequencies of magnitude 2

^q

, the corresponding dissipation term H

q

will satisfy

H

q

& min(1, 2

^2q

) L

q

.

The prefactor min(1, 2

^2q

) may be seen as a gain of two derivatives in low frequencies after time integration (like for the heat equation) whereas it corresponds to exponential decay for high frequencies. In our setting where the low and high frequencies of Z

₀

belong to the spaces B ˙

d 2−1 2,1

and B ˙

d 2+1

2,1

, respectively, we thus have k Z (t) k

^ℓ

B˙

d2−1 2,1

+ Z

_t

0

k Z k

^ℓ

B˙

d2 +1 2,1

. k Z

₀

k

^ℓ

B˙

d2−1 2,1

+ Z

_t

0

k G k

^ℓ

B˙

d2−1 2,1

, k Z(t) k

^h

˙ B

d2 +1 2,1

+ Z

t

0

k Z k

^h

˙ B

d2 +1 2,1

. k Z

0

k

^h

˙ B

d2 +1 2,1

+ Z

t

0

k G k

^h

˙ B

d2 +1 2,1

.

A rapid examination reveals that the part G

₁

of G may entail a loss of one derivative (since it is a combination of components of ∇ Z ) while G

₂

and G

₃

contain products of components of Z and Z

2

. Overcoming the difficulty with G

1

will be achieved by exploiting the symmetrizable character of the system under consideration and changing slightly the weight A ¯

₀

in the definition of L

q

for the high frequencies: we shall take

(33) L

q

, k Z

_q

k

²_L²

e A0(V)

+ 2

^−q

I

q

if q ≥ 0, with

(34) I

q

,

Z

R^d n−1

X

k=1

ε

_k

ℜ

(N M

_ω^k−1

Z c

_q

) · (N M

_ω^k

Z c

_q

) ,

where ε

₁

, · · · , ε

_n−1

> 0 will be chosen small enough (according to the Appendix).

For the low frequencies, we shall keep the original definition that we proposed in the analysis of (7), that is to say, after integrating on the whole space and using Fourier-Plancherel theorem,

(35) L

q

, k Z

_q

k

²_L²_¯

A0

+ 2

^q

I

q

if q < 0.

However, we will discover that the terms G

2

and G

3

cannot be controlled properly in the space L

¹_T

( ˙ B

d 2−1

2,1

) because Z

₂

is, somehow, too regular ! The way to overcome the difficulty is to look for an estimate of the low frequencies of the damped mode W, then to compare with Z

2

.

We shall keep in mind all the time that if choosing the coefficients ε

_k

small enough, then we have

n−1

X

k=1

ε

_k

((M

_ω

)

^t

)

^k

N

^t

N M

_ω^k−1

≤ 1 2

1 (2π)

^d

, whence, owing to Fourier-Plancherel theorem,

|I

q

| ≤ 1

2 k Z

_q

k

L²

.

Furthermore, as A ¯

0

= A

0

( ¯ V ) is definite positive and V 7→ A e

0

(V ), continuous, Condition (30) ensures that k Z

_q

k

_L²_¯

A0

≃ k Z

_q

k

_L²

and k Z

_q

k

_L²

e

A0(V)

≃ k Z

_q

k

_L²

. Therefore, we have

(36) L

q

≃ k Z

q

k

²_L²

for all q ∈ Z .

3.1.1. Basic energy estimates. The first step is devoted to studying the time evolution of k Z

q

k

²_L2

e A0(V)

and k Z

q

k

²_L2 A¯0

. The outcome is given in the following proposition.

Proposition 3.1. Let Z be a smooth solution of (3) on [0, T ] × R

^d

satisfying (29). Then, for all s ∈

_d

2

,

^d₂

+ 1

and q ≥ 0, we have:

(37) 1 2

d

dt k Z

_q

k

²_L²

e A0(V)

+ κ

₀

k Z

_2,q

k

²_L²

. k ( ∇ Z, Z

₂

) k

L^∞

k Z

_q

k

²_L²

+ c

_q

2

^−qs

k∇ Z k

_˙

B

d2 2,1

k Z k

B˙^s_2,1

k Z

_q

k

_L²

+ c

q

2

^−qs

k∇ Z k

_˙

B

d2 2,1

k Z

2

k

B˙^s−1

2,1

k Z

q

k

_L²

+ c

q

2

^−qs

k Z k

B˙^s_2,1

k Z

2

k

_˙

B

d2 2,1

+ k Z

2

k

B˙^s_2,1

k Z k

_˙

B

d2 2,1

k Z

q

k

_L²

.

Furthermore, for all s

^′

∈

_d

2

− 1,

^d₂

and q ≤ 0, we have:

(38) 1 2

d

dt k Z

_q

k

²_L²_¯

A0

+ κ

₀

k Z

_2,q

k

²_L²

. k∇ Z k

_L^∞

k Z

_q

k

²_L²

+ c

_q

2

^−qs′

k∇ Z k

_˙

B

d2 2,1

k Z k

B˙^s′

2,1

k Z

_q

k

_L²

+ c

_q

2

^−qs′

k Z k

_˙

B

d 2,12

k∇ Z k

B˙^s_2,1^′

k Z

_q

k

_L²

+ c

_q

2

^−qs′

k Z

₂

k

B˙^s_2,1^′

k Z k

_˙

B

d 2,12

k Z

_q

k

_L²

. Proof. It relies on an energy method implemented on (3) after localization in the Fourier space, and on classical commutator estimates.

In order to prove (37), apply operator ∆ ˙

_q

to (3) to get:

A e

⁰

(V )∂

_t

Z

_q

+ X

d

j=1

A e

^j

(V )∂

_j

Z

_q

+ LZ

_q

= R

¹_q

+ R

_q²

+ ˙ ∆

_q

(r(Z))

with R

¹_q

, X

d

j=1

[ A e

^j

(V ), ∆ ˙

_q

]∂

_j

Z and R

²_q

, [ A e

⁰

(V ), ∆ ˙

_q

]∂

_t

Z.

Taking the L

²

( R

^d

; R

ⁿ

) scalar product with Z

_q

, integrating by parts in the second term and using the fact that A e

^j

(V ) is symmetric yields

1 2

d dt

Z

R^d

A e

₀

(V )Z

_q

· Z

_q

+ Z

R^d

LZ

_q

· Z

_q

= 1 2

Z

R^d

∂

_t

A ˜

⁰

(V ) + X

j

∂

_j

( A e

^j

(V ))

Z

_q

· Z

_q

+ Z

R^d

(R

¹_q

+ R

²_q

) · Z

_q

+ Z

R^d

∆ ˙

_q

r(Z ) · Z

_q

. Hence, thanks to Property (12), we obtain

(39) 1 2

d

dt k Z

_q

k

²_L²

A0(Ve )

+ κ

₀

k N Z

_q

k

²_L²

≤ 1 2

Z

R^d

∂

_t

A e

⁰

(V ) + X

j

∂

_j

( A e

^j

(V ))

Z

_q

· Z

_q

+ Z

R^d

(R

¹_q

+ R

²_q

) · Z

q

+ Z

R^d

∆ ˙

q

r(Z ) · Z

q

. For the first term in the right-hand side, we have

Z

R^d

∂

_t

( A e

⁰

(V ))Z

_q

· Z

_q

. k ∂

_t

Z k

L^∞

k Z

_q

k

²_L²

. Hence, using the fact that

(40) ∂

_t

Z = ( A e

⁰

(V ))

⁻¹

H( ¯ e V + Z ) − X

d

j=1

A e

^j

(V )∂

_j

Z

,